smc-5115-68-Sigma notation

Proof, EXT2 P2 EQ-Bank 9

Using mathematical induction, show

for (3 marks)

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COMMENT: Required proof – preserve the

factor in all algebra!

Proof, EXT2 P2 EQ-Bank 7

Using mathematical induction, show

for (4 marks)

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Calculus, EXT2 C1 2022 HSC 14b

Let , where "n" is a non-negative integer.

Show that . (1 mark)

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Show that . (2 marks)

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Show that . (2 marks)

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Using parts (i) and (iii), show by mathematical induction, or otherwise, that for all ,
(2 marks)

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Using parts (ii) and (iv) prove that . (1 mark)

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i.

Mean mark (i) 93%.

ii.

♦♦ Mean mark (ii) 28%.

iii.

iv.

♦ Mean mark (iv) 50%.

♦♦ Mean mark (v) 34%.

Proof, EXT2 P2 2019 HSC 14c

Show that . (2 marks)

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Use mathematical induction to prove that, for all ,

. (2 marks)

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	$$

ii.

♦ Mean mark 45%.





	$$

Proof, EXT2 P2 2016 HSC 16c

In a group of people, each has one hat, giving a total of different hats. They place their hats on a table. Later, each person picks up a hat, not necessarily their own.

A situation in which none of the people picks up their own hat is called a derangement.

Let be the number of possible derangements.

Tom is one of the people. In some derangements Tom finds that he and one other person have each other's hat.

Show that, for , the number of such derangements is (1 mark)

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By also considering the remaining possible derangements, show that, for

(2 marks)

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Hence, show that , for (1 mark)

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Given and , deduce that , for (1 mark)

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Prove by mathematical induction, or otherwise, that for all integers (2 marks)

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♦♦♦ Mean mark 22%.

ii.

♦♦♦ Mean mark 6%.

♦♦ Mean mark part (iii) 37%.

iii.

iv.

♦♦♦ Mean mark 1%!

♦♦♦ Mean mark 13%.

Proof, EXT2 P2 2009 HSC 8a

Using the substitution $$ , or otherwise, show that

$ $ (2 marks)

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Use mathematical induction to prove that, for integers ,

$ $ (3 marks)

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Show that $$ (2 marks)

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Hence find the exact value of
(2 marks)

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$ $

ii.

	$$
	$$


	$$

$ $

	$ $
	$ $
	$ $
	$$


	$$

iii. $$

iv.

Proof, EXT2 P2 2012 HSC 16b

Show that for and . (1 mark)

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Use mathematical induction to prove

for all positive integers . (3 marks)

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Find . (1 mark)

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ii.

iii.

Proof, EXT2 P2 EQ-Bank 10

Use mathematical induction to prove that

(3 marks)

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